\newproblem{lay:4_2_31}{
  % Problem identification
	\begin{large}
	  \hspace{\fill}\newline
    \textbf{Lay, 4.2.31}
	\end{large}
	\\
  \ifthenelse{\boolean{identifyAuthor}}{\textit{Carlos Oscar Sorzano, Aug. 31st, 2013} \\}{}

  % Problem statement
	Define $T:\mathbb{P}_2\rightarrow \mathbb{R}^2$ by $T(p(t))=(p(0),p(1))$. For instance, if $p(t)=3+5t+7t^2$, then $T(p(t))=(3,15)$.
	\begin{enumerate}[a.]
		\item Show that $T$ is a linear transformation. [Hint: for arbitrary polynomials $p(t)$ and $q(t)$ in $\mathbb{P}_2$, compute
		      $T(p(t)+q(t))$ and $T(cp(t))$].
		\item Find a polynomial $p(t)$ in $\mathbb{P}_2$ that spans the kernel of $T$, and describe the range of $T$.
	\end{enumerate}
}{
  % Solution
	\begin{enumerate}[a.]
		\item Let $p(t)=a_p+b_pt+c_pt^2$ and $q(t)=a_q+b_qt+c_qt^2$. We have \\
		      \begin{center}
						$\begin{array}{rcl}
							T(p(t)+q(t))&=&T((a_p+b_pt+c_pt^2)+(a_q+b_qt+c_qt^2))\\
							   &=&T((a_p+a_q)+(b_p+b_q)t+(c_p+c_q)t^2)\\
							   &=&(a_p+a_q,a_p+a_q+b_p+b_q+c_p+c_q)\\
								 &=&(a_p,a_p+b_p+c_p)+(a_q,a_q+b_q+c_q)\\
								 &=&T(p(t))+T(q(t))
						\end{array}$\\
						$\begin{array}{rcl}
							T(cp(t))&=&T(c(a_p+b_pt+c_pt^2))\\
							   &=&T(ca_p+cb_pt+cc_pt^2)\\
								 &=&(ca_p,ca_p+cb_p+cc_p)\\
								 &=&c(a_p,a_p+b_p+c_p)\\
								 &=&cT(p(t))
						\end{array}$\\
					\end{center}
					Since $T$ meets the two conditions to be a linear transformation, it is a linear transformation.
		\item The kernel of $T$ is formed by all those polynomials such that
		      \begin{center}
						$\begin{array}{rcl}
							T(p(t))&=&(a_p,a_p+b_p+c_p)=(0,0)\\
						\end{array}$
					\end{center}
					for that we need
		      \begin{center}
						$a_p=0$\\
						$a_p+b_p+c_p=0\Rightarrow c_p=-b_p$
					\end{center}
					That is, all polynomials in the kernel of $T$ are of the form $p(t)=b_pt-b_pt^2$, in particular
		      \begin{center}
						$\mathrm{Ker}\{T\}=\mathrm{Span}\{t-t^2\}$
					\end{center}
					The range of $T$ is formed by all those vectors of the form
		      \begin{center}
						$\mathrm{Range}\{T\}=\{(a_p,a_p+b_p+c_p) \forall a_p,b_p,c_p\in\mathbb{R}\}=\mathbb{R}^2$
					\end{center}
	\end{enumerate}
}
\useproblem{lay:4_2_31}
\ifthenelse{\boolean{eachProblemInOnePage}}{\newpage}{}
